The task of this paper is to develop a Time-Domain Probe Method for the reconstructionudof impenetrable scatterers. The basic idea of the method is to use pulsesudin the time domain and the time-dependent response of the scatterer to reconstructudits location and shape. The method is based on the basic causality principle of timedependentudscattering. The method is independent of the boundary condition and isudapplicable for limited aperture scattering data.udIn particular, we discuss the reconstruction of the shape of a rough surface inudthree dimensions from time-domain measurements of the scattered field. In practise,udmeasurement data is collected where the incident field is given by a pulse. We formulateudthe time-domain fieeld reconstruction problem equivalently via frequency-domainudintegral equations or via a retarded boundary integral equation based on results ofudBamberger, Ha-Duong, Lubich. In contrast to pure frequency domain methods hereudwe use a time-domain characterization of the unknown shape for its reconstruction.udOur paper will describe the Time-Domain Probe Method and relate it to previousudfrequency-domain approaches on sampling and probe methods by Colton, Kirsch,udIkehata, Potthast, Luke, Sylvester et al. The approach significantly extends recentudwork of Chandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the timedomainudpoint source method. We provide a complete convergence analysis for theudmethod for the rough surface scattering case and provide numerical simulations andudexamples.
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